Domino tilings

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

For a positive integer n, a collection S of subsets of [n] = {1,. .. , n} is called symmetric if X ∈ S implies X * ∈ S, where X * := {i ∈ [n] : n − i + 1 / ∈ X} (the involution * was introduced by Karpman). Leclerc and Zelevinsky showed that the set of maximal strongly (resp. weakly) separated collections in 2 [n] is connected via flips, or mutations, "in the presence of six (resp. four) witnesses". We give a symmetric analog of those results, by showing that each maximal symmetric strongly (weakly) separated collection in 2 [n] can be obtained from any other one by a series of special symmetric local transformations, so-called symmetric flips. Also we establish the connectedness via symmetric flips for the class of maximal symmetric r-separated collections in 2 [n] when n, r are even (where sets A, B ⊆ [n] are called r-separated if there are no elements i 0 < i 1 < • • • < i r+1 in [n] which alternate in A \ B and B \ A). This is related to a symmetric version of higher Bruhat orders. These results are obtained as consequences of our study of related geometric objects: symmetric rhombus and combined tilings and symmetric cubillages.

Our studies are related to a special class of FASS-curves, which can be described in a node-rewriting Lindenmayer-system. These ortho-tile (or diagonal) type recursive curves inducing Hamiltonian paths. We define a special directed graph on a rectangular grid, and we enumerate all Hamiltonian paths on this graph. Our formulas are strongly related to both the Fibonacci numbers and the domino tilings of chessboards. The constructability of the regular 17-gon with straightedge and compass is also related.

The Aztec diamond of order n is the union of lattice squares in the plane intersecting the square |x| + |y| < n. The Aztec diamond theorem states that the number of domino tilings of this shape is 2 n(n+1)/2. It was first proved by Elkies, Kuperberg, Larsen and Propp in 1992. We give a new simple proof of this theorem.

is completing studies for a master's degree in mathematical sciences at Michoacán University (UMSNH). In 2013 she received a medal from the Iberoamerican Mathematics Competition for University Students. Jorge López-López received his Ph.D. at Michoacán University (UMSNH) in 2003, under the direction of Alberto Verjovsky, and has taught there since that time.

Ribbons may be used for the modeling of DNAs and proteins. The topology of a ribbon can be described by the link Lk, while its geometry is represented by the writhe Wr and the twist Tw. These three quantities are numerical integrals and are related by a single formula from knot theory. This article discusses the meanings of these three quantities, offers an approach for calculating their numerical values, and provides some examples.

This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family, the four general theorems are used to find and prove enough theorems particular for the family to explain the instability of all graphs in the family up to a certain point. A very dense family of edge-transitive unstable graphs is constructed and, at the other extreme, an unstable graph is constructed whose only symmetry is trivial. Finally, the four theorems are shown to be able to explain all instability.

Jerome Lloyd, 2, 3, ∗ Sounak Biswas, ∗ Steven H. Simon, S. A. Parameswaran, and Felix Flicker 4 Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom School of Physics and Astronomy, University of Birmingham, Edgbaston Park Road, Birmingham, B15 2TT, United Kingdom Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße, Dresden, 01187, Germany School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom

We consider the set of all tilings by dominoes (2 • 1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by a flip, i.e., a 90 ~ rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected.

In their unpublished work, Jockusch and Propp showed that a 2-enumeration of antisymmetric monotone triangles is given by a simple product formula. On the other hand, the author proved that the same formula counts the domino tilings of the quartered Aztec rectangle. In this paper, we explain this phenomenon directly by building a correspondence between the antisymmetric monotone triangles and domino tilings of the quartered Aztec rectangle.

Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the &quot;arctic circle&quot;&#39; phenomenon.However, few examples are known for which this approach works in three or more dimensions.Here we show that the rhombus tiling can be generalized to n-dimensional tiles for any $n ≥ 3$. For each $n$, we show that a certain local move is ergodic, and conjecture that it has a mixing time of $O(L^{n+2} log L)$ on regions of size $L$. For $n=3$, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron.We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions...

We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof in fact establishes a similar result for non-intersecting families of $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible &quot;combing&#39;&#39; algorithm, operating on families of paths without non-intersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.

We provide a more informal explanation of two results in our manuscript Tilings of quadriculated annuli. Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). The generating function Φ A (p, q) is such that, for q = −1, the non-zero roots in p are roots of unity and for q > 0, real negative. There is an unexpected rigidity for the −1-counting of tilings of quadriculated disks, as described in [1]: Theorem 1 Let D be a quadriculated disk. Then the determinant of the adjacency matrix of the squares of D equals-1, 0 or 1. In [4] we prove Theorem 3 below which, in a sense, extends this rigidity to annuli. In this shorter text, we present a more informal proof of this result. 1 Connectivity We assume that the squares in a quadriculated annulus A are colored in a checkerboard pattern and that the numbers of black and white squares are equal. Without loss, A is embedded in the plane.

We consider the set of all tilings by dominoes (2 1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they di er by a ip, i.e., a 90 rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region di erent from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they di er only by the order of two ips on disjoint squares: this graph is connected.

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.

Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the "arctic circle" phenomenon. However, few examples are known for which this approach works in three or more dimensions. Here we show that the rhombus tiling can be generalized to ndimensional tiles for any n ¡ 3. For each n, we show that a certain local move is ergodic, and conjecture that it has a mixing time of O ¢ L n£ 2 log L¤ on regions of size L. For n ¥ 3, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron. We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state. However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral. In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically. We conjecture that this is because the physics of the model is in a "smooth" phase where it is rigid at large scales, rather than a "rough" phase in which it is elastic.

Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the "arctic circle" phenomenon. However, few examples are known for which this approach works in three or more dimensions. Here we show that the rhombus tiling can be generalized to ndimensional tiles for any n ¡ 3. For each n, we show that a certain local move is ergodic, and conjecture that it has a mixing time of O ¢ L n£ 2 log L¤ on regions of size L. For n ¥ 3, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron. We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state. However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral. In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically. We conjecture that this is because the physics of the model is in a "smooth" phase where it is rigid at large scales, rather than a "rough" phase in which it is elastic.

We provide a combinatorial proof of the trigonometric identity cos(nθ) = T n (cos θ), where T n is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind.

We provide a combinatorial proof of the trigonometric identity cos(nθ) = T n (cos θ), where T n is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind.

In this paper, we consider domino tilings of regions of the form $\mathcal{D} \times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n \in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by $\pm 1$. We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.

Inspired by the rules of Japanese tatami layouts, we present in these paper some characterization results of tiling rectangles with 1x2 and 2x1 blocks in case a single 1x1 block is allowed. The tatami condition is that four blocks cannot meet in the same point.

Let G be a finite graph or an infinite graph on which Z Z d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Z d -action with finite quotient. The same computation also gives the entropy of the law of T.

It is known that any two domino tilings of a polygon are flip-accessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flip-accessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, we provide three

In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n ≤ 4, the flip-distance between two tilings is equal to the Hamming-distance. Conversely, for n ≥ 6, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates.

In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n ≤ 4, the flip-distance between two tilings is equal to the Hamming-distance. Conversely, for n ≥ 6, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates.

This paper studies the tilings with colored-edges triangles constructed on a triangulation of a simply connected orientable surface such that the degree of each interior vertex is even (such as, for (fundamental) example, a part of the triangular lattice of the plane). The constraints are that we only use three colors, all the colors appear in each tile and two tile can share an edge only if this edge has the same color in both tiles.

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.

The study of the dihedral f-tilings of the sphere S 2 whose prototiles are an equilateral or isosceles triangle and an isosceles trapezoid was described in . In this paper we generalize this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.

A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of planar bipartite graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions.

In this paper, we consider domino tilings of regions of the form D×[0,n], where D is a simply connected planar region and n∈N. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by ±1. We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.

We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.

We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond.

We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

The impossibility of tiling the mutilated chess board has been formalized and verified using Isabelle. The formalization is concise because it is expressed using inductive definitions. The proofs are straightforward except for some lemmas concerning finite cardinalities. This exercise is an object lesson in choosing a good formalization: one at the right level of abstraction.